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I recently asked (and then answered) this question:

https://math.stackexchange.com/questions/2756777/decreasing-sequence-of-closed-sets-in-a-separable-metric-space.

In a separable metric space there is no strictly decreasing sequence of closed sets $(X_\alpha)_{\alpha<\omega_1}$, where $X_\beta\supsetneq X_\alpha$ iff $\beta<\alpha$.

This sounds a lot like the Descending Chain Condition in Algebra. I assume it comes up frequently in topology (maybe not?). In that case, is there a name for it?

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For any topological space $X$, the following statements are easily seen to be equivalent:
(1) there is no strictly decreasing $\omega_1$-sequence of closed sets in $X$;
(2) there is no strictly increasing $\omega_1$-sequence of open sets in $X$;
(3) every open subspace of $X$ is Lindelöf;
(4) every subspace of $X$ is Lindelöf.

Spaces satisfying those equivalent conditions are called hereditarily Lindelöf spaces by some, strongly Lindelöf spaces by others.

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  • $\begingroup$ Yes, I wrote that in the comment of his answer too. Note that the dual statement (with decreasing open sets) is equivalent to hereditarily separability. Also, left and right separated sequences could be considered. It often comes up when reasoning about $L$-spaces and $S$-spaces. $\endgroup$ – Henno Brandsma Apr 29 '18 at 22:14

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